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BMO and Dirichlet Problem for Degenerate Beltrami Equation

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Abstract

Following Bojarski and Vekua, we have studied the Dirichlet problem \( \underset{z\to \zeta }{\lim}\operatorname{Re}\ f(z)=\varphi \left(\zeta \right) \) as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ(ζ) in bounded domains D of the complex plane ℂ, where f satisfies the degenerate Beltrami equation \( {f}_{\overline{z}}=\mu (z){f}_z,\left|\mu (z)\right|<1 \), a.e. in D. Assuming that D is an arbitrary simply connected domain, we have established, in terms of μ, the BMO and FMO criteria, as well as a number of other integral criteria, on the existence and representation of regular discrete open solutions to the stated above problem. We have also proven similar theorems on the existence of multivalued solutions to the problem with single-valued real parts in an arbitrary bounded domain D with no boundary component degenerated to a single point. Finally, we have given a similar solvability and representation results concerning the Dirichlet problem in such domains for the degenerate A-harmonic equation associated with the Beltrami equation.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine

    Vladimir Gutlyanskii & Vladimir Ryazanov

  2. Bogdan Khmelnytsky National University of Cherkasy, Lab. of Math. Phys., Cherkasy, Ukraine

    Vladimir Ryazanov

  3. Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine

    Evgeny Sevost’yanov

  4. Holon Institute of Technology, Holon, Israel

    Eduard Yakubov

Authors
  1. Vladimir Gutlyanskii
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  2. Vladimir Ryazanov
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  3. Evgeny Sevost’yanov
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  4. Eduard Yakubov
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Correspondence to Vladimir Gutlyanskii.

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Dedicated to the memory of Professor Uri Srebro (1936–2016)

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 19, No. 3, pp. 327–354, July–September, 2022.

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Gutlyanskii, V., Ryazanov, V., Sevost’yanov, E. et al. BMO and Dirichlet Problem for Degenerate Beltrami Equation. J Math Sci 268, 157–177 (2022). https://doi.org/10.1007/s10958-022-06189-w

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